Let X∼Gamma(α,β)X \sim \text{Gamma}(\alpha, \beta)X∼Gamma(α,β) where the PDF is f(x)=βαΓ(α)xα−1e−βxf(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}f(x)=Γ(α)βαxα−1e−βx. What is the moment generating function MX(t)M_X(t)MX(t)?
(1 - t/\beta)^{-\alpha}
(1 - \beta/t)^{-\alpha}
(1 - t/\alpha)^{-\beta}
e^{\alpha t + \beta t^2/2}